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MATHEMATICS

(311)

Time : 3 Hours ] [Maximum Marks : 100

Note (i) This Question Paper consists of two Sections, viz., ‘A’ and ‘B’

(ii) All questions from Section ‘A’ are to be attempted.

(iii) Section ‘B’ has got more than one option. Candidates are required to attempt

questions from one option only.

1. Evaluate :

2. Prove that

3. Using the properties of determinants, Prove the following :

4. Prove that

5. If r cos

6. Prove that

7. How many numbers greater than 4000 can be formed with the digits 2,3,4,5 and 6

when no digit is repeated ?

8. Determine whether the function f(x)= x3

9. Find the equation of the line which makes equal intercepts on both axes and

passes through the point (4,3).

10. Show that among rectangles of given area, the square has the least perimeter.

11. Evaluate :

12. If

13. If I, w and w2 are cube roots of unity, unity prove the following :

14. If A=

15. Find the Sum of the following series :

16. 2+22+222+…………….. to n terms

17. Find the equation of the circle which touches the axis of x and passes through the

points (-1,2) and (-3,4).

18. Find the intervals in which the function f(x)= 2x3-3x2 –12+6( a) increasing, (b)

dcreasing.

19. Find the equation of tangent to the circle x2 + y2 = 2 at the point (1,1) 4

20. Show that y =cx+a/b is the solution of the differential equation y=xdy/dx+adx/dy.

21. Solve the equation sin2 +cos= for general solution.

22. If C0, C1, C2,……, Cn are binomial coefficients in the expansion of (1+X)n ,

prove that.

23. Find the area of the region bounded by the two parabolas y2 = x and x2=y.

24. Find the sum of the following infinite series. : 1+1/3(1/2)2+ +1/5(1/2)4+……..

25. Find

26. If

27. Find the square root of 3++2i.

28. Using matrices, solve the following system of equations. : 2xy+

z=2;2x+y+z=4;x+2y+2z=5 5

29. Computer the mean and standard for the following data:

Classes 0-10 10-20 20-30 30-40 40-50 Total

Frequency 2 3 5 7 3 20

30. For events A and B, if P(A)=1/4,P(B)=1/2 P(B)=1/2, P

31. The mean and variance of binomial distribution are 3 and 4/9 respectively. Find

the distribution and P(X>1).

OPTION –II

(Linear Programming)

28. Find the dual of the following problem :

Z= 16x1+9x2+21x3

X1+x2+3x3>16

2x1+x2+x3>0

29. Four persons A,B, C and D are to be assigned four jobs I, II, III and IV. The cost

matrix is given below . Find the proper assignment :

A B C D

I

II

III

IV

8 10 17 9

3 8 5 6

10 12 11 9

6 13 9 7

30. Solve the following linear programming problem graphically :

Maximize Z = 8x+19y

Subject to

X+5y<200

2x+3y<134

x>0, y>0

OPTION –III

(Vectors and 3-D Geometry)

28. Find the Value of so that the vector and perpendicular to each other.

29. Find the equation of the line passing through the point (2,1,-4) and parallel to the

line

x+3/4=y-2/1=z-5/3

30. Find the equation of sphare concentric with the sphere x2 +y2 +z2-2x-2y-2z-6=0

This question paper consists of 36 questions [Section-A (27)+Section – B(3+3+3)]

and 8 printed pages.

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